On the path-avoidance vertex-coloring game

By Torsten Mütze and Reto Spöhel

Abstract

For any graph $F$ and any integer $r\geq 2$, the \emph{online vertex-Ramsey
density of $F$ and $r$}, denoted $m^*(F,r)$, is a parameter defined via a
deterministic two-player Ramsey-type game (Painter vs.\ Builder). This
parameter was introduced in a recent paper \cite{mrs11}, where it was shown
that the online vertex-Ramsey density determines the threshold of a similar
probabilistic one-player game (Painter vs.\ the binomial random graph
$G_{n,p}$). For a large class of graphs $F$, including cliques, cycles,
complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a
simple greedy strategy is optimal for Painter and closed formulas for
$m^*(F,r)$ are known. In this work we show that for the case where $F=P_\ell$
is a (long) path, the picture is very different. It is not hard to see that
$m^*(P_\ell,r)= 1-1/k^*(P_\ell,r)$ for an appropriately defined integer
$k^*(P_\ell,r)$, and that the greedy strategy gives a lower bound of
$k^*(P_\ell,r)\geq \ell^r$. We construct and analyze Painter strategies that
improve on this greedy lower bound by a factor polynomial in $\ell$, and we
show that no superpolynomial improvement is possible